Abstract.
We give a general result showing that the asymptotic behaviour of high moments determines the shape of distributions which are asymptotically normal. Both the factorial and non-factorial (non-central) moments are treated. This differs from the usual moment method in combinatorics, as the expected value may tend to infinity quite rapidly. Applications are given to submap counts in random planar triangulations, where we use a simple argument to asymptotically determine high moments for the number of copies of a given subtriangulation in a random 3-connected planar triangulation. Similar results are also obtained for 2-connected triangulations and quadrangulations with no multiple edges.
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References
Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation. Clarendon Press, Oxford, 1992
Bender, E.A., Gao, Z.C., Richmond, L.B.: Submaps of maps I: General 0-1 laws. J. Combin. Theory Ser. B 55, 104–117 (1992)
Billingsley, P.: Probability and Measure, 3nd ed., John Wiley & Sons, New York, 1995
Brown, W.G.: Enumeration of quadrangular dissections of the disk. Canad. J. Math. 17 (3), 302–317 (1965)
Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17–61 (1960)
Gao, Z.C., Wormald, N.C.: Sharp concentration of the number of submaps in random planar triangulations. Combinatorica 23, 467–486 (2003)
Janson, S., Luczak, T., Ruciński, A.: Random graphs, Wiley, New York, 2000
Pittel, B., Weishaar, R.S.: The random bipartite nearest neighbor graphs. Random Struct. Algorithms 15, 279–310 (1999)
Régnier, M., Szpankowski, W.: On pattern frequency occurrences in a Markovian sequence. Algorithmica 22, 631–649 (1998)
Richmond, L.B., Robinson, R.W., Wormald, N.C.: On Hamilton cycles in 3-connected cubic maps, in Cycles in Graphs (B. Alspach and C.D. Godsil, eds.) Ann. Dis. Math. 27, 141–150 (1985)
Richmond, L.B., Wormald, N.C.: Random triangulations of the plane. Eur. J. Combinatorics 9, 61–71 (1988)
Richmond, L.B., Wormald, N.C.: Almost all maps are asymmetric. J. Combin. Theory, Ser. B 63, 1–7 (1995)
Ruciński, A.: Proving normality in combinatorics. In Random Graphs Vol.2, A. Frieze and T. Luczak eds, Wiley, New York, 215–231 (1992)
Tutte, W.T.: The enumerative theory of planar maps: In A survey of combinatorial Theory, Srivastava J.N. et al. (Eds) pp. 437–448, North-Holland, Amsterdam, 1973
Acknowledgments.
The authors would like to thank an anonymous referee for pointing out how to significantly shorten the proof of Theorems 1 and 2 and incidentally remove the condition σ n ( log 2σ n )=o(μ n ). We would also like to thank Boris Pittel for some suggestions.
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Revised version: 6 February 2004
Research supported by NSERCC and University of Macau.
Research supported by the Australian Research Council and the Canada Research Chairs program. Research carried mainly while the author was at the Department of Mathematics and Statistics, University of Melbourne.
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Gao, Z., Wormald, N. Asymptotic normality determined by high moments, and submap counts of random maps. Probab. Theory Relat. Fields 130, 368–376 (2004). https://doi.org/10.1007/s00440-004-0356-9
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DOI: https://doi.org/10.1007/s00440-004-0356-9